Sine and Cosine are Periodic on Reals/Sine/Proof 2

Theorem

The real sine function is periodic with the same period as the real cosine function.

Proof

Since Real Cosine Function is Periodic, let $L$ be its period.

From Primitive of Cosine Function:

$\ds \int \cos x \rd x = \sin x + C$

for any constant $C$.

Therefore $\sin x$ is a Primitive of $\cos x$, for the special case of $C = 0$.

From Primitive of Periodic Real Function, it follows that $\sin x$ is periodic with period $L$.

$\blacksquare$